Is there anyone can tell me any information about the integer solution to the combinatotial equation $$ \sum (-1)^k \binom{n}{k} \alpha_k = b_n $$ (all variables are integers)? For example, suppose alpha_{0}=0 , when n=2, it is 2alpha_{1}-alpha_{1}=-b_{2} If we take b_{2} to be a given number, this is a first degree Diophantine Equation, we know how to solve it using elementary number theory, right? But when n=3, take b_{3} to be a given number, alpha_{0}=0, could you write down a general solution to this Dioph. equation? Thank you very much!